Neural Data Compression

Lossless bit reduction with machine learning by minimizing cross-entropy. Examples: NNCP and TRACE models.
  • data compression means encoding into less bits
  • lossless means without loosing any information
  • trade-off between communication throughput, computation, and memory
  • can compress if some symbols are more likely than others
  • better symbol prediction => lower cross-entropy => higher compression
Neural Data Compression
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Dictionary Coder

Morse Code Uses Huffman Coding Compression

  • compresses 26 characters of english alphabet, not compressing white space
  • character mapped to sequence of dots and dashes, space mapped to space
  • more frequent characters mapped to fewer dots and dashes
  • this is Huffman coding with a static tree
  • encoding and decoding requires minimal compute and memory
A part of Morse Huffman tree
A part of Morse Huffman tree

GZip’s Deflate Data Compression

  • GZip uses Deflate compression format (ZIP also)
    1. a sliding widow of 32k bytes is used to detect duplicate strings
      • duplicate strings are referenced back with length and distance symbols (similar to dictionary coding)
      • this along with byte literals defines custom alphabet of symbols
    2. Huffman coding maps frequent symbols to shorter bit sequences
      • bit-mapping trees stored in the output and sometimes refreshed
Deflate algorithm illustration with LZ77 and Huffman coding
Deflate algorithm illustration with LZ77 and Huffman coding

Arithmetic Coding vs Huffman Coding

  • arithmetic coding has higher compression ratio than Huffman, but slower
  • maps symbol of probability \( q \) to length \( -log_2 q \) in contrast to Huffman
  • defined by split of \( (0, 1) \) into subintervals of the probability size, sorted by the size.
  • encodings are numbers within the subintervals in binary format
  • transmit enough digits so all fractions that fall within interval (prefix code)
aritmetic coding interval visualization
aritmetic coding interval visualization

Entropy and Cross-Entropy in Compression

  • Let true next symbol probability given previous symbols: \( p(x \mid x_i, x_{i-1}, …) \)
    • estimated next symbol probability given previous symbols: \( q(x \mid x_i ,… ) \)
    • arithmetic coding encodes to length \( - \log_2 q(x) \)
  • then average compressed message bit-length is cross-entropy: \( - \sum_x p(x) \log_2 q(x) \)
  • Cross-Entropy minimization equals likelihood maximization: \( \frac{1}{N} \log ( \prod_k q_k^{N_k p_k} ) \)

Bits-per-byte (bpb) and Bits-per-Character (bpc)

  • compression ratio is defined as \( \mathrm{cmpRatio} = \mathrm{unCompressedBytes} / \mathrm{compressedBytes} \)
  • Bits-per-byte is defined as \( \mathrm{compressedBits} / \mathrm{unCompressedBytes} \)
  • Bits-per-byte (bpb) metric is inverse compression ratio divided by 8: \( 1 bpb = 1 / (8 \mathrm{cmpRatio}) \).
  • Bits-per-character (bpc) metric for ASCII Extended characters equals bits-per-byte (bpb).
  • Cross-entropy loss (log2) for a character-level language model averaged over a dataset equals bpc.
  • Perplexity is cross-entropy (log2) to the second power \( PP = 2^{\mathrm{crossEnropy}} \)
  • Gzip compresses enwik8 2.92 bpb, Morse code approximately 10.8 bpc
  • SRU++ model achieves 1.02 bpc - approximately compression ratio of 8

Compression by Predicting Next Symbol

  • Huffman coding predicts next symbol cheaply with symbol frequency
  • can trade more memory and computation with complex probability modeling with ML
  • ML model can be trained on already compressed data stream deterministically
  • common benchmarks are enwik8, and enwik9 datasets with bits-per-byte (bpb)
  • bpb not comparable to language modeling results: single-pass, extra overhead, compressing entire dataset
model predicting the next symbol from alphabet
model predicting the next symbol from alphabet

Language Modelling is Compression

NNCP: Lossless Data Compression with Neural Networks

  • model is not stored in the output. It is deterministically derived based on decompressed output
  • model is regularly retrained during compression
  • creates a tokenization dictionary (16k symbols like Cmix) during the first pass
  • tokenization is dictionary coding
  • NNCP 1 uses multi-layer LSTM to predict next symbol
  • NNCP v2 Transformer beats Cmix on enwik9

NNCP v2 Results vs Cmix

  • results of NNCP v2 on enwik9 below. LSTM (large2) is …
  • 10,000 times slower than GZip for 2.8x less bits-per-byte
  • faster, simpler, better than Cmix (complex w/ LSTM) on enwik9
  • worse than Cmix on enwik8
NNCP v2, CMIX, LSTM compression performance
NNCP v2, CMIX, LSTM compression performance

TRACE Model 1-layer Transformer

  • TRACE is 1-layer transformer compression
  • predicts the next byte instead of dictionary symbol (token)
  • vocabulary size is 256 too little compared to the hidden dimension
    • so 4 consecutive embeddings concatenated before input
  • retrained less often (adaptive) than NNCP, but starts randomly initialized
TRACE model architecture
TRACE model architecture

TRACE Model Results Faster Data Compression Than NNCP

  • 3x speedup with competitive compression with NNCP, but still 1000x slower than GZip
  • worse on text but on-par on other due to not using tokenization dictionary
  • result on enwik9 below for Cmix, NNCP, Dzip required GPU
TRACE, NNCP, CMIX, Dzip compression performance
TRACE, NNCP, CMIX, Dzip compression performance

Deep Neural Network Lossless Compression Applications

  • other compression algos: Cmix (HutterPrice 2021 SoTA), Dzip
  • as of 2022-05 seems unpractical - slow, small compression improvement
    • could be practical as a side effect of other computation
  • note that lossy compression of images and video seem more likely applied
  • more overview of the field in this paper

Abstraction is Lossy Compression

Autoencoders and Variational Autoencoders are Lossy Compressors

  • if finite precision encoding into less latent dimensions
  • compressed representations may be used for interpolation
  • interpretable features may be disentangled

Created on 14 May 2022. Updated on: 03 Nov 2023.
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